Energy-Time Uncertainty Problem.

1. The problem statement, all variables and given/known data
Following a question that asks for the Energy-Time uncertainty principle, [tex]\Delta E \Delta T \geq \frac{\hbar}{2}[/tex]

Show that for any wave, the fractional uncertainty in wavelength, [tex]\frac{\Delta \lambda}{\lambda}[/tex] is the same in magnitude as the fractional uncertainty in frequency [tex]\frac{\Delta f}{f}[/tex]

Now I thought I might be able to do a similar thing for wavelength using [tex]E = \frac{hv}{\lambda}[/tex] but this runs into problems stemming from the fact that lambda is the denominator of that equation not the numerator. I have tried several rearrangements and even considered using Position-Momentum Uncertainty with De Broglie, but I still can't get see how I can equate the fractional uncertainties. I'm not even sure if my rearrangement for the frequency is on the right track or not.

[itex]v = f\lambda[/itex] applies for any wave, where [itex]v[/itex] is the speed of the wave. The formula for photons is just a special case.

Yeah I mentioned trying to use [tex]E=\frac{hv}{\lambda}[/tex] In my original post, I have been trying it for several hours now and still can't seem to make any progress, my main issue is that whilst something like [tex] \Delta E = h \Delta f [/tex] is true, the similar statement, [tex] \Delta t = \frac{1}{\Delta f} [/tex] is not true. Hmm... will try some more.

Wait, surely one of derivations is wrong, and I have a feeling it is the one in terms of wavelength. The reason is because if I assume my working to be true, then we know that somehow we can equate the proportional uncertainty in frequency and wavelength. But from my workings thus far that would also mean that the proportional uncertainty in energy is equivalent to the proportional uncertainty in time. Which I don't think is necessarily the case, as they are linked only by the uncertainty principle which I don't think implies such a relation.

I should note that although this is posted in the homework section, it is not really homework but a revision problem for exams, and therefore I will gain no marks from its solution nor am I ever likely to be given the solution by my lecturers. So any help on the problem would be greatly appreciated. I chose to post it in the homework section as it is a university question and not a general discussion, but I worry that this was perhaps the wrong thing to do if people are reticent to help me for fear of helping someone to cheat.

Any kind of wave is a periodic travelling disturbance. Consider plane waves travelling in the x direction. The functional form is f(t/T-x/L) where T is the time period and L is the period in space, the wavelength. The speed of wave is that of a "wave-front" the place where the wave has a specified property, for example a maximum. For a wavefront, t/T-x/L = const. ---> x=L/T*t +constant, the wavefront travels with the speed v= L/T. As T is the reciprocal of frequency, v=L*f. The speed of wave is determined by the kind of wave and the medium it travels, its uncertainty is zero:

Hmm... the minus sign seems to mess it up as wouldn't that go to i? But the partial derivative method seems to get very close to the solution you gave and I can see why you would express the uncertainty as the sum of the two products via such a method.

But v has no uncertainty, it is specified for the kind of wave. [tex]\Delta v =0 \rightarrow f \Delta L = -L \Delta f[/tex]

The same method can be applied for the derivation of an implicit function.

For example, you have a circle x2+y2=R2=25, and you want to find the slope of its tangent at the point (3,4)
Instead of expressing y with x, and taking the derivative with respect to x, you can consider R2 as function of x and y and find the differential of R2 which is obviously zero.
[tex] d(R^2)=2x\cdot dx +2y\cdot dy=0[/tex]
From here, the differential quotient is dy/dx = -x/y, the slope of the tangent is -3/4. It looks very formal, but is is correct and works!

In error calculus, x and y are independent variables and their variances add up to give the variance of a function f(x,y).
In our case, L and f are not independent.